MA1035: Engineering Modeling using Dynamic Systems

Session 07: Eigenfunctions for Systems of LODEs

Prof. Alejandro Ucan-Puc

Tecnológico de Monterrey • Departamento de Ciencias

Goals

  • Use eigenvalues and eigenvectors to solve systems of ODEs.

  • Use the Wronskian to test for linear independence.

Conventions for Systems of ODEs

In what follows we will deal with systems of two equations with two unknown functions whose ODEs are linear with constant coefficients. This is described in the general form $$\begin{array}{c} x'= a_{11}x+a_{22}y+f_1(t) \\ y'=a_{21}x+a_{22}y+f_2(t) \end{array}$$ where $a_{ij}$ are constants and $f_1(t)$ and $f_2(t)$ are functions of $t.$

Solving for Homogeneous systems

Given the system $$X'=AX$$ consider the matrix $A.$ Suppose that $(\lambda,v)$ is an eigenvalue-eigenvector pair of $A$ and assume that $\lambda$ is real. Let $F=e^{\lambda t}v$ be a vector of functions, then the vector $F$ is a solution vector of the system.

Example:

Consider the system $$X'=AX$$ where $$A=\left(\begin{array}{cc} 3 & 1 \\ 0 & -3 \end{array}\right).$$ Find the eigenvalues and eigenvectors of $A$ and construct the general solution.

Example:

Consider the system $$X'=\left(\begin{array}{cc} 1 & 2 \\ -2 & -3 \end{array}\right) X.$$ Find the eigenvalues and eigenvectors of $A$ and construct the general solution.

Test for linear independence

Definition: Consider the solution function vectors $X_1$ and $X_2$ for a system of ODEs. The Wronskian of the solutions is the determinant of the matrix $$\left(\begin{array}{cc} x_1 & x_2 \\ y_1 & y_2\end{array}\right)$$ and is denoted by $W(X_1,X_2).$



Theorem [Abel's]: The solutions $X_1$ y $X_2$ are linearly independents if $$W(X_1,X_2)\neq 0.$$

Solution for Systems of ODEs

Method of Eigenfunctions: For the system $X'=AX.$

  1. Find the eigenvalues and eigenvectors of $A.$
  • If the eigenvalues are real and distinct, then the general solution $$X_1=e^{\lambda_1}v_1\mbox{ and } X_2=e^{\lambda_2}v_2.$$
  • If the eigenvalues are real and repeated, then the general solution $$X_1=e^{\lambda}v_1\mbox{ and } X_2=te^{\lambda}v_2.$$
  • If the eigenvalues are complex conjugated, then the general solution $$X_1=e^{\alpha t}\left(\cos(\beta t)v_1+\sin(\beta t)v_2\right)\mbox{ and } X_2=e^{\alpha t}\left(\cos(\beta t)v_1-\sin(\beta t)v_2\right).$$
  1. Express the general solution as $$X=c_1 X_1+c_2 X_2.$$
  2. If there are initial conditions, use them to find the constants $c_1$ and $c_2.$

Example 1:

Find the general solution for $X'=AX$ if $A=\left(\begin{array}{cc} 1 & 2 \\ 4 & 3 \end{array}\right).$

  • $A=\left(\begin{array}{cc} -4 & 2 \\ -\frac{5}{2} & 2 \end{array}\right).$

Example 2:

Find the general solution for $X'=AX$ if $A=\left(\begin{array}{cc} 10 & -5 \\ 8 & -12 \end{array}\right).$

Example 3:

Find the general solution for $A=\left(\begin{array}{cc} 3 & -1 \\ 9 & -3 \end{array}\right).$

Summary & Next Step

Eigenvalues, eigenvectors, characteristic equations det(A - lambda I) = 0, fundamental set of solutions, and linear independence.

Next Session: Generalized Eigenvectors & Complex Eigenvalues